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The haversine formula is a very accurate way of computing distances between two points on the surface of a sphere using the latitude and longitude of the two points. The haversine formula is a re-formulation of the spherical law of cosines, but the formulation in terms of haversines is more useful for small angles and distances.
"Sticking 90,0 and -90,0 in there I get:-
haversine( [ 90, 0 ], [ -90 , 0 ] )
Distance: 20015086.796 m
Distance: 20015.087 kmWhich is half the circumference of the Earth. (If it was using straight line distance it would give me the diameter of the Earth).
Haversine should give you great circle distance, it won't be a direct straight line between the two points.